State Variable Constraints Research Articles

Aircraft Turbofan Engine Health Estimation Using Constrained Kalman Filtering

Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically. For instance, state-variable constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. This paper develops an analytic method of incorporating state-variable inequality constraints in the Kalman filter. The resultant filter is a combination of a standard Kalman filter and a quadratic programming problem. The incorporation of state-variable constraints increases the computational effort of the filter but significantly improves its estimation accuracy. The improvement is proven theoretically and shown via simulation results obtained from application to a turbofan engine model. This model contains 16 state variables, 12 measurements, and 8 component health parameters. It is shown that the new algorithms provide improved performance in this example over unconstrained Kalman filtering.

Receding Horizon Control for Legged Robot–Slack Variable Method for Swing Leg

AbstractThe slack variable method is introduced into receding horizon control with inequality constraint, and formulation about swing leg is described. Generally, the conditions that swing leg does not fall under the floor altitude can be described as inequality state variable constraint. It is difficult to treat this as a states variable constraint, because there is not control input variable explicitly. In this paper, it proposes dealing this with a problem using slack variable. This simple formulation can be applied to a varied legged robot with any number of legs. © 2004 Wiley Periodicals, Inc.

Discrete model predictive controller design using Laguerre functions

In Model Predictive Controller (MPC) design, the traditional approach of expanding the future control signal uses the forward shift operator to obtain the linear-in-the-parameters relation for predicted output. As a consequence, in case of rapid sampling, complicated process dynamics and/or high demands on closed-loop performance, satisfactory approximation of the control signal requires a very large number of forward shift operators, and leads to poorly numerically conditioned solutions and heavy computational load when implemented on-line. In this paper, by using a performance specification on the exponential change rate of the control signal, a more appropriate expansion, related to Laguerre net-works, is introduced and analyzed. It is shown that the number of terms used in the optimization procedure can be reduced to a fraction of that required by the usual procedure. By relaxing the constraint on the exponential change rate of the control signal and allowing arbitrary complexity in describing the trajectory, the proposed approach becomes equivalent to the traditional approach in MPC design. Closed-loop stability of the proposed model predictive control system is analyzed by using terminal state variable constraints.

Normal Vectors on Manifolds of Critical Points for Parametric Robustness of Equilibrium Solutions of ODE Systems

{Equilibrium solutions of systems of parameterized ordinary differential equations \dot x = f(x, α) , x ∈ Rn , α∈ Rm can be characterized by their parametric distance to manifolds of critical solutions at which the behavior of the system changes qualitatively. Critical points of interest are bifurcation points and points at which state variable constraints or output constraints are violated. We use normal vectors on manifolds of critical points to measure the distance between these manifolds and equilibrium solutions as suggested in I. Dobson [J. Nonlinear Sci., 3:307-327, 1993], where systems of equations to calculate normal vectors on codimension-1 bifurcations were presented. We present a scheme to derive systems of equations to calculate normal vectors on manifolds of critical points which (i) generalizes to bifurcations of arbitrary codimension, (ii) can be applied to state variable constraints and output constraints, (iii) implies that the normal vector defining system of equations is of size c1 n+ c2 m+ c3 , ci∈ R , i.e., no bilinear terms nm or higher-order terms occur, (iv) reduces the number of equations for normal vectors on Hopf bifurcation manifolds compared to previous work, and (v) simplifies the proof of regularity of the normal vector system. As an application of this scheme, we present systems of equations for normal vectors to manifolds of output/state variable constraints, to manifolds of saddle-node, Hopf, cusp, and isola bifurcations, and we give illustrative examples of their use in engineering applications.}

Optimal Control of One-Dimensional Partial Differential Algebraic Equations with Applications

We present an approach to compute optimal control functions in dynamic models based on one-dimensional partial differential algebraic equations (PDAE). By using the method of lines, the PDAE is transformed into a large system of usually stiff ordinary differential algebraic equations and integrated by standard methods. The resulting nonlinear programming problem is solved by the sequential quadratic programming code NLPQL. Optimal control functions are approximated by piecewise constant, piecewise linear or bang-bang functions. Three different types of cost functions can be formulated. The underlying model structure is quite flexible. We allow break points for model changes, disjoint integration areas with respect to spatial variable, arbitrary boundary and transition conditions, coupled ordinary and algebraic differential equations, algebraic equations in time and space variables, and dynamic constraints for control and state variables. The PDAE is discretized by difference formulae, polynomial approximations with arbitrary degrees, and by special update formulae in case of hyperbolic equations. Two application problems are outlined in detail. We present a model for optimal control of transdermal diffusion of drugs, where the diffusion speed is controlled by an electric field, and a model for the optimal control of the input feed of an acetylene reactor given in form of a distributed parameter system.

An extended model for pairwise conflict resolution in air traffic management

A conflict resolution model for aircraft traversing planar intersecting trajectories is developed and solved. The model incorporates the flight dynamics of each aircraft. Three types of optimal conflict avoidance manoeuvres are examined. The first is the minimum-time safe deviation to a designated objective and the second and third are the minimum-time safe deviation to and from a designated ground track. These objectives lead to singular optimal control problems where the control variable constraint is defined in terms of the maximum allowable turn rate ∣u(t)∣⩽uM,0⩽t⩽tf, for the aircraft executing the avoidance manoeuvre and the non-autonomous state variable constraint is defined in terms of the radius rp of the protected zone about the potentially conflicting aircraft. A fast, accurate procedure is derived for computing the optimal steering program which requires the solution of a coupled differential-algebraic system of equations to determine the control law on any boundary arc of an extremal trajectory. An example pairwise trajectory conflict is analysed in detail. The incorporation of variable airspeed v(t) and an acceleration/deceleration control u2(t) is also discussed. Copyright © 1999 John Wiley & Sons Ltd.

The optimal control of collision avoidance trajectories in air traffic management

A collision avoidance/conflict resolution model for aircraft traversing planar intersecting trajectories is developed based on the actual flight dynamics of each aircraft and the criteria that a flight regime correction is optimal if it represents a minimum-time safe deviation from a preassigned flight program. The formulation of the model leads to a singular optimal control problem where the control variable constraint is the maximum acceptable aircraft turn rate for passenger comfort and safety and the state variable constraint is defined in terms of the radius of the protected zone about a potentially conflicting aircraft. A robust, accurate real-time solution procedure is derived for computing the steering program for the execution of the optimal safe avoidance maneuver. Several representative examples of pairwise trajectory conflicts are analysed in detail. A brief discussion of how the model might be generalized to incorporate changes in airspeed and altitude as well as more comprehensive dynamics is included.

Dynamic Process Optimization through Adjoint Formulations and Constraint Aggregation

A dynamic optimization strategy is developed that makes use of existing differential−algebraic equation solvers and nonlinear programming strategies. In contrast to current dynamic optimization approaches, this approach applies an adjoint strategy for the calculation of objective function and constraint gradients. Furthermore, we consider two aggregation approaches, the Kreisselmeier−Steinhauser function and the smoothed penalty function, for state variable constraints. The advantage of this aggregation is to reduce the burden in the calculation of the adjoint system. The resulting implementation has a significant advantage over current strategies that rely on the solution of sensitivity equations, especially for large dynamic optimization problems. Performance of this approach is demonstrated on several dynamic optimization problems in process engineering.

Dynamic optimization with state variable path constraints

A new method for solving dynamic optimization problems that contain path constraints on the state variables is described. We establish the equivalence between the inequality path-constrained dynamic optimization problem and a hybrid discrete/continuous dynamic optimization problem that contains switching phenomena. The control parameterization method for solving dynamic optimization problems, which transforms the dynamic optimization problem into a finite-dimensional nonlinear program (NLP), is combined with an algorithm for constrained dynamic simulation so that any admissible combination of the control parameters produces an initial value problem that is feasible with respect to the path constraints. We show that the dynamic model, which is in general described by a system of differential-algebraic equations (DAEs), can become high-index during the state-constrained portions of the trajectory. During these constrained portions of the trajectory, a subset of the control variables are allowed to be determined by the solution of the high-index DAE. The algorithm proceeds by detecting activation and deactivation of the constraints during the solution of the initial value problem, and solving the resulting high-index DAEs and their related sensitivity systems using the method of dummy derivatives. This method is applicable to a large class of dynamic optimization problems.

Time optimal seek control for a coupled dual stage disk drive actuator

A unified expression and three typical kinds of seeking trajectories of the time optimal control of a dual stage disk drive actuator with passive coupling between the two stages are presented. Necessary conditions for finding the optimal control for the problem with state-variable constraints are applied to derive the solution. Numerical calculations and computer simulations were performed to get the trajectories. The experimental trajectories of 1/3 full stroke for a disk drive are also presented. A DSP (Digital Signal Processor) based control system is used to implement the optimal control. The time optimal seek control features cooperative control of the two stages of the actuators, and seek time can decrease as compared with one-stage seeking when the fine actuator locked onto the coarse actuator.

Dynamic optimization of an industrial semi‐batch nylon 6 reactor with end point constraints and stopping conditions

In this study, optimal vapor release rate (or pressure) histories have been generated for an industrial semi-batch nylon 6 reactor using Pontryagin's minimum principle. The batch time has been taken as the objective function, which is to be minimized. The pressure is constrained to lie between a lower and an upper limit. The temperature, a state variable, is also constrained to lie between 220°C and 270°C in order to ensure single-phase polymerization. Optimization has been carried out with a single end-point constraint (on monomer conversion) and a stopping condition (obtaining a product having a desired degree of polymerization, μn). Techniques have been developed to overcome the discontinuities present in the model, as well as to take care of state variable constraints. The effects of various physical and computational variables on the optimal pressure history and the corresponding batch time have been studied. It is found that the optimal batch time is almost 50% of the industrial value used currently. Interestingly, the optimal pressure history is quite similar qualitatively with the current practice though quantitatively there is a significant difference. Improvements in reactor operation along these lines have been reported. © 1996 John Wiley & Sons, Inc.

Dynamic optimization of an industrial semi-batch nylon 6 reactor with end point constraints and stopping conditions

In this study, optimal vapor release rate (or pressure) histories have been generated for an industrial semi-batch nylon 6 reactor using Pontryagin's minimum principle. The batch time has been taken as the objective function, which is to be minimized. The pressure is constrained to lie between a lower and an upper limit. The temperature, a state variable, is also constrained to lie between 220°C and 270°C in order to ensure single-phase polymerization. Optimization has been carried out with a single end-point constraint (on monomer conversion) and a stopping condition (obtaining a product having a desired degree of polymerization, μn). Techniques have been developed to overcome the discontinuities present in the model, as well as to take care of state variable constraints. The effects of various physical and computational variables on the optimal pressure history and the corresponding batch time have been studied. It is found that the optimal batch time is almost 50% of the industrial value used currently. Interestingly, the optimal pressure history is quite similar qualitatively with the current practice though quantitatively there is a significant difference. Improvements in reactor operation along these lines have been reported. © 1996 John Wiley & Sons, Inc.

Seek time and trajectories of time optimal control for a dual stage optical disk drive actuator

A piecewise expression of the seek time and three kinds of trajectories with their seek distance conditions of the time optimal control of a dual stage disk drive actuator are presented. Necessary conditions for finding the optimal control for the problem with state-variable constraints are applied to derive the solution. The seek time can decrease when the two stages of the actuator are cooperatively controlled as compared with that when the fine actuator locked onto the coarse actuator.

SIMULTANEOUS DESIGN OF CONTROL AND STRUCTURAL SYSTEMS

Abstract A numerical algorithm for simultaneous optimization of control and structural systems is proposed, analyzed and evaluated in this paper. In the procedure, the structural and control design subproblems are solved in a sequence. In the outer loop, it is proposed to solve the constrained structural optimization problem using the nonlinear programming ( NLP) approach, and in the inner loop, the control design problem is proposed to be solved using the differential dynamic programming ( DDP) approach. For the optimal control design problem, the functional and pointwise state variable constraints are treated using the augmented Lagrangian approach. The resulting unconstrained control design subproblem is then solved using the DDP approach which has been determined to be more efficient than the NLP approach for this class of problems. The basic ideas of the augmented Lagrangian and the DDP approaches are explained. The proposed approach is quite general as it can treat linear as well as nonlinear system...

A direct application of higher‐order parametric programming techniques to structural optimization

AbstractComputational methods based on a sequence of parametric programming problems are presented for solving constrained optimization problems (COP) without any parameter. An auxiliary parametric programming problem (APPP) is formulated in order to solve COP. The procedure is started with an arbitrary initial solution which is the trivial solution of APPP corresponding to the initial value of the parameter. Then the optimal solution for the final value of the parameter, which is the optimal solution of COP, is estimated by Taylor's expansion with respect to the parameter where higher‐order terms are incorporated. It is shown that the incorporation of the higher‐order terms indeed leads to a faster convergence of the solution. As an extension of the method, a general algorithm is presented for optimum design problems with state variable constraints which are implicit functions of the design variables. Logarithmic penalty functions are incorporated and the weight coefficients for the penalty terms are updated continuously. The derivatives of the state variables with respect to the parameter and their sensitivity coefficients are expressed explicitly in terms of those of the design variables. Finally, a method of simultaneous analysis and optimization is developed for trusses with geometrical non‐linearity.

Second-Order Multiplier Update Calculations for Optimal Control Problems and Related Large Scale Nonlinear Programs

A second-order multiplier update rule is applied to K-stage discrete-time optimal control problems with control and state variable constraints. Each update entails the assembly and solution of sparse block-banded equilibrium equations. Several direct elimination solution techniques are considered, and it is shown that the updates can always be calculated in $O( K )$ flops. Part of the analysis applies not only to control problems, but also to other similarly structured large scale nonlinear programs with equality and inequality constraints.

Decomposition strategies for large-scale dynamic optimization problems

Abstract Recently, efficient strategies have been developed to solve dynamic simulation and optimization problems in a simultaneous manner. These rely on the ability to obtain an accurate algebraic discretization of the differential equations as well as the ability to solve large optimization problems in an efficient manner. These concerns have been addressed by applying orthogonal collocation on finite elements to these systems and solving the nonlinear program (NLP) with a reduced space successive quadratic programming (SQP) approach. In a recent study we discussed theoretical properties of these differential algebraic equation (DAE) systems and cautioned that application of orthogonal collocation may not yield a stable discretization nor an accurate solution to the control problem. As a result of this, preanalysis of the DAE system is required and appropriate approximation error criteria must be embedded within the nonlinear program. In this paper we tailor this approach to the accurate solution of optimal control problems. The optimal control problem has a natural partitioning of control variables and state variables for the NLP. Note here that partitioned spaces are not orthogonal. We develop a decomposition strategy to: (1) exploit the block matrix form of the discretized differential equations which results from using collocation on finite elements, and (2) allow us to perform the optimization in the control space. Here the state variables for each finite element are determined by linearized differential equations, and a coordination step is used to update the control variables and integration length. Information is passed from element to element by chainruling the state information. While the approach has much in common with earlier quasilinearization approaches, the nonlinear programming strategy has a great deal of flexibility in determining control variable discontinuities, enforcing a wide variety of state and control variable constraints and ensuring the accurate determination of both state and control variable profiles. Two classes of problems are investigated; first we consider problems where the differential equations are linear in the state variables and then we consider the general nonlinear (states and controls) problem. Example problems are illustrated for both classes of problems.

A constrained pseudo-newton control strategy for nonlinear systems

The duality of control theory and the theory on the solution of operator equations has been exploited to develop a nonlinear control law. The control law is developed by analogy to Newton's method and several related properties are shown. To handle the state and control variable constraints, the method is generalized to include a successive quadratic programming (SQP) algorithm. Stability properties for this algorithm have been studied. It was found that with large enough but finite sampling time, systems that are open-loop asymptotically stable in the large can approach the setpoint monotonically. Simulation results of two example problems demonstrate the effectiveness of the proposed strategy.

Simultaneous optimization and solution methods for batch reactor control profiles

Differential-algebraic optimization problems appear frequently in process engineering, especially in process control, reactor design and process identification applications. For fed-batch reactor systems the optimal control problem is especially difficult because of the presence of singular arcs and state variable constraints. For problems of this type we propose a simultaneous optimization and solution strategy based on successive quadratic programming (SQP) and orthogonal collocation on finite elements. In solving the resulting nonlinear programming (NLP) problem, a number of interesting analogs can be drawn to more traditional methods based on variational calculus. First, the collocation method has very desirable stability and accuracy properties. Secondly, it will be shown that NLP optimality conditions have direct parallels to general variational conditions for optimal control. To demonstrate this strategy, we consider the optimization of a fed-batch penicillin reactor using a number of cases. For the simplest case, the results presented here agree well with previously obtained, analytically-based solutions. In addition, accurate results are presented for more difficult cases where no analytic solution is available.

Optimal temperature control for batch beer fermentation

Optimal control theory was applied to the process of batch beer fermentation. The performance functional considered was a weighted sum of maximum ethanol production and minimum time. Calculations were based on the model of Engasser et al. modified to include temperature effects. Model parameters were determined from isothermal batch fermentations. The fermentor cooling duty was the single available control. Temperature state variable constraints as well as control variable constraints were considered. The optimal control law is shown to be bang-bang control with the existence of a singular arc corresponding to isothermal operation at the maximum temperature constraint. An iterative algorithm is presented for computing appropriate switching times using a penalty-function-augmented performance functional.